WORKING WITH AN ONLINE GAME AS AN ENTRY POINT TO ALGEBRAIC THINKING
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12th International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
WORKING WITH AN ONLINE GAME AS AN ENTRY POINT TO
ALGEBRAIC THINKING
Angeliki Kolovou and Marja van den Heuvel-Panhuizen
FISME, Utrecht University, the Netherlands
a.kolovou@uu.nl, m.vandenheuvel-panhuizen@uu.nl
Olaf Köller
IPN Leibniz Institute, Kiel, Germany
koeller@ipn.uni-kiel.de
In an experimental study we investigated whether an intervention including an online game
contributed to primary school students’ performance on problems with covarying quantities. In total
236 sixth graders participated in the study. The students in the experimental group were requested to
solve a number of problems by playing the game at home. Special software recorded students’ online
activity. The results of a written pre and posttest on problem solving showed a significant positive
effect of the intervention on posttest performance. Moreover, playing the game stimulated the
students to generate a general rule for the relationship between covarying quantities.
Early algebra; Online game; Information and Communication Technology; Primary school
INTRODUCTION
The integration of algebraic reasoning into the primary school mathematics curriculum has
received growing attention from researchers and policy makers (Cai & Knuth, 2011; Garraher
& Schliemann, 2007; Katz, 2007; NCTM, 2000). Given the significance of algebraic
reasoning as an educational goal as well as students’ difficulties with algebra in secondary
school, primary school should better prepare students for the study of algebra in the later
grades. However, this does not mean adding formal algebra to the primary school
mathematics curriculum, but providing students with entry points to algebra (Carraher,
Schliemann, Brizuella & Enrnest, 2006; Kaput, 2008) through tasks that offer them
opportunities for reasoning algebraically in a context-connected, informal way.
The present study investigated how primary school students’ in grade 6 can be provided with
opportunities to develop algebraic reasoning, in particular the ability to solve contextual
number problems with covarying quantities, hereafter called early algebra performance. A
dynamic computer game, containing animations and providing instant feedback, was
designed to offer students experiences of covariation and functional relations; in this
computer environment students can observe how output values vary as input values change,
so that they can recognize relations between covarying quantities. Our first research question
was whether an intervention including an online game has an effect on students’ performance
abcdeLast names of authors, in order on the paper in early algebra, while our second question zoomed in on the effect of specific characteristics of online work. THEORETICAL BACKGROUND Teaching and learning of early algebra According to Carraher et al. (2006) early algebra involves a shift from working with particular numbers and measures towards working with relations among sets of numbers and measures, especially functional relationships. Moreover, Kaput (2008) considers functional thinking as one of the core strands of algebraic reasoning. Along this line, our approach on early algebra focuses on supporting students to reason about relationships between numbers and quantities and to express general rules for the relationship between covarying quantities. A task that can prompt such reasoning is, for example, the Quiz problem: In a quiz you get 2 points each time an answer is correct. In case a question is not answered or the answer is false, 1 point is subtracted from your score. The quiz contains 10 questions. Tina received 8 points in total. How many questions did Tina answer correctly? The Quiz problem can be resolved by applying a formal algebraic approach or by reasoning informally about the relationships between the numbers in the problem. In the latter case, students can solve this problem as follows: “If all 10 questions were correctly answered, then I would get 20 points. Every wrong answer means in total 3 points less (missing 2 points, because of missing a correct answer and missing 1 point, because of getting a penalty point for the wrong answer). To have 8 points left, means that there were four wrong answers (four times 3 points subtracted from the 20 points). So, there are six correctly answered questions.” Although this approach cannot be labeled as a formal algebraic procedure, Johanning (2004) suggests that that such informal reasoning can be seen as a way in which students make sense of algebraic situations. Role of ICT in teaching and learning algebra Several studies have emphasized the role of the Information and Communication Technology (ICT) in the teaching and learning of mathematics (e.g., Li & Ma, 2010; Slavin & Lake, 2008) and especially in the teaching and learning of algebra (e.g., Rojano & Sutherland, 1994; Lannin, 2005). Cuoco (1995) showed that dynamic computer environments can support the development of increasingly sophisticated concepts of functions. Resnick, Eisenberg, Berg, Mikhak, and Willow (2000) also suggested that technology tools might be more appropriate for developing concepts related to dynamic processes, such as the concept of function. Among the ICT environments, computer games have attracted special interest from educators, researchers, and policymakers (McFarlane, Sparrowhawk, & Heald, 2002). On the basis of an extensive literature review, Mitchell and Savill-Smith (2004) asserted that computer games have a significant impact on students’ cognitive skills. They concluded that computer games are engaging and can embed mathematical concepts that may be hard to grasp with concrete materials. However, apart from a few studies that show positive effects on students’ performance (Kebritchi, Hirumi, & Bai, 2010; Redfield, Gaither & Redfield, 2009), less is known about whether games can contribute to the learning of algebra. Abcde+3 ICME-12, 2012
Last names of authors in order as on the paper METHOD Design – Procedure A pretest-posttest-control-group experiment was set up to investigate the influence of an intervention, including playing an online game, on students’ early algebra performance. The intervention consisted of three periods of one week in which the students in the experimental group received three sets of problems with the request to solve them at home by using the online environment and present their solutions in whole class discussions. Before and after the intervention a pretest and a posttest were administered in the experimental as well as in the control group. However, the students in the control group did not receive the intervention. Sample The sample consisted of 236 sixth-grade students from ten schools (five experimental and five matching control schools) covering a wide socio-economic range. All participating students were receiving a similar type of mathematics education and had no early algebraic experiences. Also, they were familiar with playing computer games at school and at home. With respect to age and gender the experimental and the control group were rather similar. Also, no significant difference in the mean pretest scores of the two groups was found (t = 1.19, n.s.). However, we found a significant difference (t = 3.36, p = .001, d = .48) in their general mathematical ability as measured by the Grade 5 CITO-LOVS Mathematics Test1 (CITO E5). Pre- and posttest Early algebra problem solving was assessed by a written test containing seven items, which where contextual number problems with covarying quantities, such as the Quiz problem. After excluding one item due to poor psychometric properties (M = .17, item-total-correlation r = .37), the internal consistency of the test was satisfying (Cronbach’s alpha = .79). The online environment A dynamic game called ‘Hit the target’2 was developed to give students experience in dealing with covarying quantities (Figure 1). In this game the students can set the shooting mode (user or computer shooting) and the game rule mode (user or computer defined). The features of the game are dynamically linked. In the course of the game the values on the scoreboard update rapidly to provide information about the total score. In this way, students may become aware of the fact that the arrows, the score, and the game rule are related to each other so that a modification in the value of one of these variables has a direct effect on the other variables. Moreover, the game offers instant visual feedback by displaying the number of hits and misses and the number of points resulting from a shooting action. ICME-12, 2012 abcde+2
Last names of authors, in order on the paper
Figure 1a: Screen view of game in the user Figure 1b: Screen view of game in the
shooting mode computer shooting mode
The series of problems that the students of the experimental group were asked to solve in the
online environment varied from finding the number of hits and the number of misses that
produce a particular score, to generating a general solution. A selection of these problems is
the following:
Problem 3: What is the game rule to get 15 points in total with 15 hits and 15 misses? Are there
other game rules to get 15 hits, 15 misses, and 15 points?
Problem 4: What is the game rule to get 16 points in total with 16 hits and 16 misses? Are there
other game rules to get 16 hits, 16 misses, and 16 points?
Problem 5: What is the game rule to get 100 points in total with 100 hits and 100 misses? Are there
other game rules to get 100 hits, 100 misses, and 100 points? Can you explain your answer?
To keep track of students’ online activity, the game was connected to the Digital Mathematics
Environment (DME).3 The log data (see Figure 2) consist of a list of the actions performed by
a student per session (i.e., every time a student logged in) and per event (i.e., every time a
student clicked the shoot button) and the logged-in time. Other characteristics of the online
activity were the number of problems that the students worked on and the number and
percentage of focused events (i.e., shooting actions intended to answer a given problem).
RESULTS
To answer the first question we compared the performance of the experimental versus the
control group. We found that in the latter group no change occurred between pre- and posttest
(MPRE = .31, SD = .30, MPOST = .33, SD = .34), whereas students in the experimental group
showed significant increase with medium effect size in their achievement (MPRE = .35,
SD = .35, MPOST = .48, SD = .37, t = −5.69, p = .000, d = .57). Moreover, a regression analysis
revealed that the effect of the intervention was significant4 (B = .076, p = .006, see Table 1).
Abcde+3 ICME-12, 2012Last names of authors in order as on the paper
Table 1: Regression model predicting posttest scores by CITO E5, pretest, and group
B SE p
CITO E5 .013 .002 < .001
Pretest score .596 .063 < .001
Group .076 .028 .006
R2(explained variance) = .71
Because playing the game was not compulsory, some students of the experimental group did
not go online and were only involved in class discussions. To disentangle the effect of the
online working from that of the class discussions, we compared the performance of the
students who followed these discussions and were logged in (n = 96) and the students who
only followed the class discussions (n = 27). The two groups did not differ significantly in
their pretest performance (t = 1.50, n.s.) and CITO E5 performance (t = .38, n.s.). With
respect to the posttest performance, a regression analysis with CITO E5 score, pretest score
and group (0 = only-class-discussions, 1 = logged-in) as predictors revealed that the two
groups did not differ significantly (B = .026, n.s.).
To investigate the influence of specific characteristics of the online working on the problem
solving performance in early algebra, we focused on the group of students who went online
(the logged-in group) and examined their log files (Figure 2). Because the characteristics
(logged-in time, number of events, number of focused events, percentage of focused events,
and number of worked problems) were strongly correlated, we performed a principal
component analysis (PCA). As a result, all characteristics collapsed into a single factor that
we considered as an indicator of effort. The regression analysis on the posttest scores with
CITO E5 score, pretest score, and effort as predictor variables revealed that the effect of effort
put in the online work was not significant (B = .028, n.s.).
In addition to the quantitative results, the students’ log files provided a glimpse on the
potential of the game to stimulate algebraic thinking. Figure 2 shows how a students’ online
activity led to the discovery of the general rule for acquiring an equal number of hits, misses
and total points. Table 2 summarizes her focused events and is extended with the score gained
from each shooting action.
ICME-12, 2012 abcde+2Last names of authors, in order on the paper
session: 1 date: 2008/11/17 04:13:58 duration: 00:05:59 total events: 5
...
...
event: 3
who shoots: computer hits: 15 misses: 15 at-random: 0
game rule: student hits: 4 added misses: 2 added
Problem 3
event: 4
who shoots: computer hits: 15 misses: 15 at-random: 0
game rule: student hits: 1 added misses: 1 added
event: 5
who shoots: student hits: 1 misses: 0 at-random: 6 (hits: 1 misses: 5)
game rule: student hits: 1 added misses: 1 added
session: 2 date: 2008/11/17 04:20:43 duration: 00:02:52 total events: 2
event: 1
who shoots: student hits: 1 misses: 0 at-random: 1 (misses: 1)
game rule: student hits: 0 less misses: 0 less
event: 2
who schoots: computer hits: 15 misses: 15 at-random: 0
game rule: student hits: 5 added misses: 4 less Problem 3
session: 3 date: 2008/12/01 04:02:24 duration: 00:36:03 total events: 7
event: 1
who shoots: computer hits: 16 misses: 16 at-random: 0
game rule: student hits: 2 added misses: 1 less
event: 2
who shoots: computer hits: 16 misses: 16 at-random: 0 Problem 4
game rule: student hits: 3 added misses: 2 less
event: 3
who shoots: computer hits: 16 misses: 16 at-random: 0
game rule: student hits: 100 added misses: 99 less
event: 4
who shoots: computer hits: 10 misses: 10 at-random: 0
game rule: student hits: 2 added misses: 1 less
Problem 5
event: 5
who shoots: computer hits: 10 misses: 10 at-random: 0
game rule: student hits: 1000 added misses: 999 less
event: 6
who shoots: student hits: 3 misses: 0 at-random: 0
game rule: student hits: 50 added misses: 49 less
event: 7
who shoots: computer hits: 10 misses: 10 at-random: 0
Problem 5
game rule: student hits: 50 added misses: 49 less
Figure 2: Part of student’s log file
Abcde+3 ICME-12, 2012Last names of authors in order as on the paper
Table 2: Student’s focused events
Problem Session Event Hits Misses Game rule Score
Points Points
per per
hit miss
Problem 3: 15 h-15 m-15 pa 1 3 15 15 +4 +2 90
4 15 15 +1 +1 30
2 2 15 15 +5 −4 15
Problem 4: 16 h-16 m-16 p 3 1 16 16 +2 −1 16
2 16 16 +3 −2 16
3 16 16 +100 −99 16
Problem 5:100 h-100 m-100 p 4 10 10 +2 −1 10
5 10 10 +1000 −999 10
7 10 10 +50 −49 10
a
h-m-p stands for hits-misses-points
First, the student found one solution in Problem 3 (+5 points for a hit and –4 points for a miss)
by trial-and-error. Next, the student applied the general rule (i.e., the difference between the
points for a hit and for a miss should be 1) to solve Problems 4 and 5. This understanding was
evoked by working on the series of problems, by which the student could experience that at a
more general level the solution to Problem 3 was also applicable to Problems 4 and 5.
Coming to Problem 5 she showed clearly that the size of the numbers does not matter. She
chose not only for +1000 (for a hit) and –999 (for a miss) but she did fill in 10 hits and 10
misses instead of the required 100 hits and 100 misses. In this way she showed to be aware of
the fact that what works for 100 arrows also works for 10 arrows. Her actions reveal that her
reasoning was not anymore bound to the specific values and reflect algebraic thinking instead
of applying an arithmetical procedure.
DISCUSSION
Students benefitted from the intervention. After controlling for the differences in the pretest
and the general mathematics achievement, the students in the experimental group performed
significantly higher than the students in the control group, even though not all students from
the experimental group went online. Actually, because of its voluntary character, the fact that
some students did not go online is an inherent feature of the intervention. This situation may
ICME-12, 2012 abcde+2Last names of authors, in order on the paper also occur when teachers carry out this intervention in their educational practice in the future. In fact, the question we aimed to answer in this study was whether the intervention, in any degree of implementation, had an effect or not. Yet, it was somewhat surprising that the performance of the logged-in students and the students who only attended the class discussions did not differ significantly. However, this result might not be very reliable because the number of students who did not logged in was rather small. The analysis of the influence of the students’ online work on their performance resulted in another noteworthy outcome. Effort did not predict students’ posttest performance. It might be that students already benefited from the game after a short period of playing the game. Furthermore, there may be a critical threshold of intervention implementation, above which increased implementation does not meaningfully influence outcomes (Durlak & DuPre, 2008). Also, we should keep in mind that students may not exclusively rely on the game to work on the problems, but they may solve them partly in their head. Therefore, the effort of the online work might not completely capture students’ cognitive effort related to the gain in performance. In general, the findings of our study indicate that computers are suitable as a tool to improve students’ performance in early algebra. Moreover, home computing may create an effective learning environment supporting and extending school learning. Furthermore, the positive effects on learning suggest that teachers can be more comfortable with letting students taking responsibility for participating in voluntary computer activities carried out at home. However, some limitations should be kept in mind when interpreting the results of our study. The effect of the intervention was examined with only one game and the operationalization of early algebra focused on one type of problems. In addition, early algebra competency was measured by a test consisting of a limited number of items. Another limitation of the study might be that the items were too difficult for the students, which might have obscured the effects of the intervention. Nevertheless, making the items easier would not have been appropriate, because lowering the cognitive demand of the items would undermine their algebraic character. A further limitation is that the control group did not get an alternative intervention with an online computer game on another topic. In this way, it is difficult to separate the real effect of the intervention from increased motivation caused by the so-called Hawthorn effect (Parsons, 1974). However, the main goal of our study was to investigate whether an intervention including an online game has an effect on students’ early algebra performance. Nonetheless, further research is necessary to disclose what aspects of the online work did contribute to the increase in performance and disentangle cognitive and motivational effects. Finally, we have to take into account that the duration of the intervention was quite short. A longer intervention might have resulted into stronger effects. Nevertheless, the significant results of this short intervention are an indication of its power. Overall, the promising results of our study encourage us to continue this line of research and further pursue the development of algebraic reasoning in the primary school through computer games. Especially, the effect of an intervention including compulsory participation Abcde+3 ICME-12, 2012
Last names of authors in order as on the paper in class should be explored. In fact, students might be able to gain more from the intervention if the online activity is carried out by all students at school under the teacher’s supervision. Notes 1. The CITO-LOVS Mathematics Test is a series of standardized tests for monitoring Dutch primary school students’ mathematics performance. 2. The game was developed by the second author and programmed by Huub Nilwick at the Freudenthal Institute. 3. The DME is developed by Peter Boon at the Freudenthal Institute. 4. Because in the regression analyses the standard errors may be underestimated due to the nested structure of the data, we employ a .01 alpha criterion of significance. References Cai, J., & Knuth, E. (2011). Early algebraization. A global dialogue from multiple perspectives. Berlin Heidelberg: Springer-Verlag. Carraher, D.W., & Schliemann, A.D. (2007). Early Algebra and Algebraic Reasoning. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (Vol. 2, pp. 669– 705). Charlotte, NC: Information Age Publishing. Carraher, D.W., Schlieman, A.D., Brizuella, B.M., & Enrnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115. Cuoco, A. (1995). Computational media to support the learning and use of functions. In A. diSessa, C. Hoyles, & R. Noss (Eds.), Computers and exploratory learning (pp. 79–108). Berlin: Springer. Durlak, J.A., & DuPre, E.P. (2008). Implementation matters: A review of research on the influence of implementation on program outcomes and the factors affecting implementation. American Journal of Community Psychology, 41, 327–350. Johanning, D.I. (2004). Supporting the development of algebraic thinking in middle school: A closer look at students’ informal strategies. Journal of Mathematical Behavior, 23, 371–388. Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades (pp. 5–18). New York: Lawrence Erlbaum. Katz, V.J. (Ed.). (2007). Algebra: Gateway to a technological future. Washington, DC: Mathematical Association of America. Kebritchi, M., Hirumi, A., & Bai, H. (2010). The effects of modern mathematics computer games on mathematics achievement and class motivation. Computers & Education, 55(2), 427-443. Lannin, J. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. Li, Q., & Ma, X. (2010). A Meta-analysis of the Effects of Computer Technology on School Students’ Mathematics Learning. Educational Psychology Review, 22, 215–243. McFarlane, A., Sparrowhawk, A., & Heald, Y. (2002). Report on the educational use of games. Cambridge, United Kingdom: TEEM ICME-12, 2012 abcde+2
Last names of authors, in order on the paper Mitchell, A., & Savill-Smith, C. (2004). The use of computer and video games for learning: A review of the literature. London: Learning and Skills Development Agency. National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Parsons, H. M. (1974). What happened at Hawthorne? Science, 183(4128), 922-932. Redfield, C., Gaither, D., Redfield, N. (2009). COTS Computer Game Effectiveness. In R. Ferdig (Ed.), Handbook of Research on Effective Electronic Gaming in Education (pp. 277-294). Hershey, PA: Information Science Reference/IGI Global. Resnick, M., Eisenberg, M., Berg, R., Mikhak, B., & Willow, D. (2000). Learning with Digital Manipulatives: New Frameworks to Help Elementary-School Students Explore "Advanced" Mathematical and Scientific Concepts. Proposal to the National Science Foundation. Rojano, T. & Sutherland, R. (1994). Towards an algebraic notion of function: the role of spreadsheets. In D. Kirshner (Ed.), Proceedings of the 16th annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 278– 284). Baton Rouge, LA: Louisiana State University. Slavin, R., & Lake, C. (2008). Effective programs in elementary mathematics; A best-evidence synthesis. Review of Educational Research, 78 (3), 427-515. Abcde+3 ICME-12, 2012
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